However, it requires to transmit \(\mathbf {x}_i^k\)  with infinite precision communication. To reduce communication cost by only transmitting \(\mathsf {Q}(\mathbf {x}_i^k)\) , Ref. [1]} proposes \(X^{k+1}=W\mathsf {Q}(X^k)\) , which was then improved in [2]} as \(X^{k+1}=X^k+(W-I)\mathsf {Q}(X^k)\) . Since \(\mathbf {x}_i^k\)  does not converge to zero, we cannot expect the compression error \(\mathsf {Q}(\mathbf {x}_i^k)-\mathbf {x}_i^k\)  to vanish. Thus, both algorithms cannot converge to \(\bar{\mathbf {x}}\) .
